Per unit quantities, Bus admittance matrix

 

PER-UNIT QUANTITIES (PU SYSTEM) – GATE LEVEL NOTES

1. Definition

Per-unit value is the ratio of actual value to base value.

$$\text{Per-unit value} = \frac{\text{Actual value}}{\text{Base value}}$$

It is a normalized quantity widely used in power system analysis.

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2. Base Quantities

Four base quantities exist, but only two are independent:

• Base Power → $S_{base}$
• Base Voltage → $V_{base}$
• Base Current → $I_{base}$
• Base Impedance → $Z_{base}$

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3. Base Relationships

Base Current

Single phase:

$$I_{base} = \frac{S_{base}}{V_{base}}$$

Three phase:

$$I_{base} = \frac{S_{base}}{\sqrt{3} V_{base}}$$

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Base Impedance

Single phase:

$$Z_{base} = \frac{V_{base}^2}{S_{base}}$$

Three phase:

$$Z_{base} = \frac{V_{base}^2}{S_{base}}$$

(Same formula for both)

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Base Admittance

$$Y_{base} = \frac{1}{Z_{base}}$$

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4. Per-Unit Conversion Formulas

Voltage:

$$V_{pu} = \frac{V_{actual}}{V_{base}}$$

Current:

$$I_{pu} = \frac{I_{actual}}{I_{base}}$$

Impedance:

$$Z_{pu} = \frac{Z_{actual}}{Z_{base}}$$

Power:

$$S_{pu} = \frac{S_{actual}}{S_{base}}$$

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5. Change of Base Formula (VERY IMPORTANT FOR GATE)

$$Z_{pu,new} = Z_{pu,old} \times \frac{S_{base,new}}{S_{base,old}} \times \left( \frac{V_{base,old}}{V_{base,new}} \right)^2$$

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6. Advantages of Per-Unit System

1. Simplifies calculations

Transformer turns ratio disappears.

2. Same impedance value on both sides of transformer

3. Avoids unit conversions

4. Numerical values are within small range (0–1 pu)

5. Easy fault analysis

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7. Typical Per-Unit Values (GATE FAVORITE)

Generator reactance:
0.1 – 0.3 pu

Transformer reactance:
0.05 – 0.15 pu

Transmission line reactance:
0.2 – 0.5 pu

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8. Transformer Per-Unit Impedance Property (VERY IMPORTANT)

Per-unit impedance is same on both sides:

$$Z_{pu,primary} = Z_{pu,secondary}$$

(if same base MVA is used)

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BUS ADMITTANCE MATRIX (Ybus) – GATE LEVEL NOTES

1. Definition

Bus admittance matrix relates bus current and bus voltage:

$$[I] = [Y_{bus}] [V]$$

Where:

• $I$ = bus current vector
• $V$ = bus voltage vector
• $Y_{bus}$ = bus admittance matrix

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2. General Form

For n-bus system:

$$\begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} Y_{11} & Y_{12} & Y_{13} \\ Y_{21} & Y_{22} & Y_{23} \\ Y_{31} & Y_{32} & Y_{33} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \$$

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3. Formation Rules of Ybus (MOST IMPORTANT FOR GATE)

Diagonal Elements

$$Y_{ii} = \text{Sum of admittances connected to bus i}$$

Includes:

• Line admittance
• Shunt admittance

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Off-Diagonal Elements

$$Y_{ij} = - (\text{Admittance between bus i and j})$$

Negative sign always.

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4. Example (2 Bus System)

Line impedance:

$$Z = j0.5$$

Admittance:

$$Y = \frac{1}{j0.5} = -j2$$

Ybus matrix:

$$Y_{bus} = \begin{bmatrix} -j2 & +j2 \\ +j2 & -j2 \end{bmatrix}$$

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5. Properties of Ybus Matrix (IMPORTANT)

Property 1: Square matrix

n buses → n × n matrix

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Property 2: Symmetrical matrix

$$Y_{ij} = Y_{ji}$$

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Property 3: Diagonal elements positive magnitude

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Property 4: Sparse matrix

Most elements are zero.

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Property 5: Sum of each row = 0 (if no shunt)

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6. Methods of Formation

Method 1: Inspection Method (GATE FAVORITE)

Directly use rules.

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Method 2: Singular Transformation Method

Uses incidence matrix.

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7. Ybus with Shunt Admittance

If shunt admittance exists:

Add it only to diagonal element.

Example:

Bus 1 has shunt admittance Ysh

$$Y_{11} = (\text{line admittance}) + Y_{sh}$$

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8. Importance of Ybus in Power System

Used in:

Load flow analysis

Fault analysis

Stability analysis

Power flow studies

Short circuit analysis

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GATE EXAM IMPORTANT POINTS (VERY HIGH PROBABILITY)

Remember:

Diagonal element:
Sum of all connected admittances

Off diagonal:
Negative of admittance between buses

Per-unit impedance:

$$Z_{base} = \frac{V_{base}^2}{S_{base}}$$

Change of base formula (very important)

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ONE-LINE REVISION SHEET

Per-unit value:
Actual / Base

Base impedance:

$$Z_{base} = \frac{V_{base}^2}{S_{base}}$$

Ybus equation:

$$I = Y_{bus} V$$

Diagonal element:
Sum of admittances

Off diagonal:
Negative of branch admittance